A Neuro-Evolution Heuristic Using Active-Set Techniques to Solve a Novel Nonlinear Singular Prediction Differential Model

Zulqurnain Sabir,Wajaree Weera,Thongchai Botmart,Muhammad Asif Zahoor Raja

doi:10.3390/fractalfract6010029

Abstract

In this study, a novel design of a second kind of nonlinear Lane–Emden prediction differential singular model (NLE-PDSM) is presented. The numerical solutions of this model were investigated via a neuro-evolution computing intelligent solver using artificial neural networks (ANNs) optimized by global and local search genetic algorithms (GAs) and the active-set method (ASM), i.e., ANN-GAASM. The novel NLE-PDSM was derived from the standard LE and the PDSM along with the details of singular points, prediction terms and shape factors. The modeling strength of ANN was implemented to create a merit function based on the second kind of NLE-PDSM using the mean squared error, and optimization was performed through the GAASM. The corroboration, validation and excellence of the ANN-GAASM for three distinct problems were established through relative studies from exact solutions on the basis of stability, convergence and robustness. Furthermore, explanations through statistical investigations confirmed the worth of the proposed scheme.

Highlights

Along with the details of singular points, prediction terms and shape factors
The detail of the numerical results for and discussions of three different problems based on the second kind of NLE-PDSM using the designed artificial neural networks (ANNs)-GAASM is provided
In order to find the precision and accuracy of the novel designed second kind of NLE-PDSM, three problems involving trigonometric and hyperbolic trigonometric functions represented with the second kind of prediction differential equation were designed, and numerical investigations were accomplished by combining artificial neural networks with global and local search proficiencies via the genetic algorithm and the active-set method

Summary

Introduction

Along with the details of singular points, prediction terms and shape factors. The modeling strength of ANN was implemented to create a merit function based on the second kind of NLE-PDSM using the mean squared error, and optimization was performed through the GAASM. It is always difficult to solve LE models due to a singular point at the origin, and there are few existing deterministic methods that have been implemented to solve the singular models [10,11,12,13,14]. Where Ω ≥ 1 is the value of the shape vector, f is dependent on t, g is some known function of dependent variable f (t), and t = 0 shows the singularity at the origin.

Objectives

Results

Conclusion